Contents
Chapter 1. Introduction 1
Chapter 2. The Brauer group and the Tate-Shafarevich group 13
1. Brauer groups and
(9x-gerbes
13
2. Tate-Shafarevich groups and genus one fibrations 28
3. Complementary fibrations 33
Chapter 3. Smooth genus one fibrations 43
1. Gx-gerbes 43
2. The class of the lifting gerbe 47
3. The class of the extension gerbe 49
4. Duality between the lifting and extension presentations 53
Chapter 4. Surfaces 63
1. The lifting presentation 63
2. The extension presentation 65
3. Duality for gerby genus one fibered surfaces 67
Chapter 5. Modified T-duality and the SYZ conjecture 79
Appendix A. Duality for representations of 1-motives,
by Dmitry Arinkin 83
Bibliography 87
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